A139548 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-1 + x + 2\,n\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} + 16\,n + 32\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	24	32	8																											64
2	320	1024	1152	512	64																									3072
3	3584	18432	38400	40960	23040	6144	512																							131072
4	36864	262144	802816	1376256	1433600	917504	344064	65536	4096																					5242880
5	360448	3276800	13271040	31457280	48168960	49545216	34406400	15728640	4423680	655360	32768																			201326592
6	3407872	37748736	190316544	576716800	1167851520	1660944384	1695547392	1245708288	648806400	230686720	51904512	6291456	262144																	7516192768
7	31457280	411041792	2480930816	9160359936	23091740672	41984983040	56679727104	57579405312	44084232192	25190989824	10496245760	3053453312	572522496	58720256	2097152															274877906944
8	285212672	4294967296	30198988800	131533373440	396948930560	879394553856	1477871403008	1919313510400	1943304929280	1535450808320	940463620096	439697276928	152672665600	37580963840	6039797760	536870912	16777216													9895604649984
9	2550136832	43486543872	349100310528	1752346656768	6160593715200	16099684909056	32391032733696	51256139710464	64604092760064	65256659353600	52857894076416	34170759806976	17441325318144	6899864961024	2053531238400	438086664192	61605937152	4831838208	134217728											351843720888320
10	22548578304	429496729600	3876207984640	22033182228480	88438745333760	266356691828736	624273496473600	1165310526750720	1758370348400640	2164148121108480	2182182688784384	1803456767590400	1217333318123520	665891729571840	291327631687680	99883759435776	26011395686400	4896262717440	612032839680	42949672960	1073741824									12384898975268864
11	197568495616	4157528342528	41669772705792	264569985433600	1193872059269120	4071732075823104	10895653425119232	23439388880928768	41202050767257600	59819273706536960	72210408974319616	72715376869244928	61101115285962752	42728052647526400	24721230460354560	11719694440464384	4486445527990272	1357244025274368	314176857702400	52913997086720	5952824672256	377957122048	8589934592							432345564227567616
12	1717986918400	39582418599936	436231238320128	3059940860100608	15334476355731456	58417052783738880	175737967124414464	428113543972257792	859200098666545152	1437615851363631104	2021647290980106240	2401471933527883776	2415766409322692608	2058404514452471808	1482541346718744576	898509907102269440	454870640470523904	190272686209892352	64745566835310592	17525115835121664	3651065798983680	556352883654656	56899726737408	3298534883328	68719476736					14987979559889010688
13	14843406974976	371634930188288	4466765987840000	34304762786611200	189033536605388800	795584623626158080	2657975901660119040	7232587487510528000	16318525518695628800	30919311509107507200	49642672367400386560	67960022064462561280	79640650856792064000	80049064450929459200	69021897409219788800	50970016548346920960	32121729178906132480	17177395282837504000	7729827877276876800	2893034995004211200	885991967220039680	216977624625315840	41094247088128000	5717460464435200	536011918540800	28587302322176	549755813888			516508834063867445248
14	127543348822016	3448068464705536	44886462692327424	374608009629794304	2251250057871360000	10373760266671226880	38109160979646382080	114564185802166763520	287061397379292856320	607537348950884352000	1096604914856346255360	1699999963700747304960	2274536988469981347840	2634757778499163914240	2646520090010320896000	2305413056186768424960	1739351814712338677760	1133333309133831536640	634876529653674147840	303768674475442176000	123026313162554081280	41659703928060641280	11598440298153246720	2593440066667806720	450250011574272000	57632001481506816	4987384743591936	246290604621824	4398046511104	17708874310761169551360

Technical details

Node id: n-858b0d55747d5753 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c94e3f620e523ccf

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:          1;
  1:         24,          32,            8;
  2:        320,        1024,         1152,           512,            64;
  3:       3584,       18432,        38400,         40960,         23040,           6144,            512;
  4:      36864,      262144,       802816,       1376256,       1433600,         917504,         344064,          65536,           4096;
  5:     360448,     3276800,     13271040,      31457280,      48168960,       49545216,       34406400,       15728640,        4423680,         655360,          32768;
  6:    3407872,    37748736,    190316544,     576716800,    1167851520,     1660944384,     1695547392,     1245708288,      648806400,      230686720,       51904512,        6291456,         262144;
  7:   31457280,   411041792,   2480930816,    9160359936,   23091740672,    41984983040,    56679727104,    57579405312,    44084232192,    25190989824,    10496245760,     3053453312,      572522496,      58720256,       2097152;
  8:  285212672,  4294967296,  30198988800,  131533373440,  396948930560,   879394553856,  1477871403008,  1919313510400,  1943304929280,  1535450808320,   940463620096,   439697276928,   152672665600,   37580963840,    6039797760,    536870912,    16777216;
  9: 2550136832, 43486543872, 349100310528, 1752346656768, 6160593715200, 16099684909056, 32391032733696, 51256139710464, 64604092760064, 65256659353600, 52857894076416, 34170759806976, 17441325318144, 6899864961024, 2053531238400, 438086664192, 61605937152, 4831838208, 134217728;

%F Formula

The row polynomials satisfy (2*n-1+x)*P(n,x) = (16*n+8+(32*n+24)*x+(16*n+24)*x^2+8*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 8 + (32*n + 24)*x + (16*n + 24)*x^2 + 8*x^3)*P[n - 1, x])/(2*n - 1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 8 + (32*n + 24)*x + (16*n + 24)*x^2 + 8*x^3)*v[i - 1])/(2*n - 1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    
    ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.